# NAG Library Routine Document

## 1Purpose

c06pwf computes the two-dimensional inverse discrete Fourier transform of a bivariate Hermitian sequence of complex data values.

## 2Specification

Fortran Interface
 Subroutine c06pwf ( m, n, y, x,
 Integer, Intent (In) :: m, n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Out) :: x(m*n) Complex (Kind=nag_wp), Intent (In) :: y((m/2+1)*n)
#include <nagmk26.h>
 void c06pwf_ (const Integer *m, const Integer *n, const Complex y[], double x[], Integer *ifail)

## 3Description

c06pwf computes the two-dimensional inverse discrete Fourier transform of a bivariate Hermitian sequence of complex data values ${z}_{{j}_{1}{j}_{2}}$, for ${j}_{1}=0,1,\dots ,m-1$ and ${j}_{2}=0,1,\dots ,n-1$.
The discrete Fourier transform is here defined by
 $x^ k1 k2 = 1mn ∑ j1=0 m-1 ∑ j2=0 n-1 z j1 j2 × exp 2πi j1 k1 m + j2 k2 n ,$
where ${k}_{1}=0,1,\dots ,m-1$ and ${k}_{2}=0,1,\dots ,n-1$. (Note the scale factor of $\frac{1}{\sqrt{mn}}$ in this definition.)
Because the input data satisfies conjugate symmetry (i.e., ${z}_{{j}_{1}{j}_{2}}$ is the complex conjugate of ${z}_{\left(m-{j}_{1}\right)\left(n-{j}_{2}\right)}$, the transformed values ${\stackrel{^}{x}}_{{k}_{1}{k}_{2}}$ are real.
A call of c06pvf followed by a call of c06pwf will restore the original data.
This routine calls c06pqf and c06prf to perform multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in Brigham (1974) and Temperton (1983).

## 4References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350

## 5Arguments

1:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the first dimension of the transform.
Constraint: ${\mathbf{m}}\ge 1$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the second dimension of the transform.
Constraint: ${\mathbf{n}}\ge 1$.
3:     $\mathbf{y}\left(\left({\mathbf{m}}/2+1\right)×{\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayInput
On entry: the Hermitian sequence of complex input dataset $z$, where ${z}_{{j}_{1}{j}_{2}}$ is stored in ${\mathbf{y}}\left({j}_{2}×\left(m/2+1\right)+{j}_{1}\right)$, for ${j}_{1}=0,1,\dots ,m/2$ and ${j}_{2}=0,1,\dots ,n-1$. That is, if y is regarded as a two-dimensional array of dimension $\left(0:{\mathbf{m}}/2,0:{\mathbf{n}}-1\right)$, ${\mathbf{y}}\left({j}_{1},{j}_{2}\right)$ must contain ${z}_{{j}_{1}{j}_{2}}$.
4:     $\mathbf{x}\left({\mathbf{m}}×{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the real output dataset $\stackrel{^}{x}$, where ${\stackrel{^}{x}}_{{k}_{1}{k}_{2}}$ is stored in ${\mathbf{x}}\left({k}_{2}×m+{k}_{1}\right)$, for ${k}_{1}=0,1,\dots ,m-1$ and ${k}_{2}=0,1,\dots ,n-1$. That is, if x is regarded as a two-dimensional array of dimension $\left(0:{\mathbf{m}}-1,0:{\mathbf{n}}-1\right)$, ${\mathbf{x}}\left({k}_{1},{k}_{2}\right)$ contains ${\stackrel{^}{x}}_{{k}_{1}{k}_{2}}$.
5:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=3$
An internal error has occurred in this routine. Check the routine call and any array sizes. If the call is correct then please contact NAG for assistance.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

Some indication of accuracy can be obtained by performing a forward transform using c06pvf and a backward transform using c06pwf, and comparing the results with the original sequence (in exact arithmetic they would be identical).

## 8Parallelism and Performance

c06pwf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pwf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken by c06pwf is approximately proportional to $mn\mathrm{log}\left(mn\right)$, but also depends on the factors of $m$ and $n$. c06pwf is fastest if the only prime factors of $m$ and $n$ are $2$, $3$ and $5$, and is particularly slow if $m$ or $n$ is a large prime, or has large prime factors.
Workspace is internally allocated by c06pwf. The total size of these arrays is approximately proportional to $mn$.